Multiple Yang–Mills solutions in every topological component

Let $\mathcal{A}_k(A_0)$ denote the components of the space of connections with boundary value $A_0$, and let $(\mathcal{D}_{\epsilon})$ be the Dirichlet problem for Yang–Mills connections with coupling parameter $\epsilon>0$. For example, in $\mathcal{A}_0(A_0)$ one may consider connections of the form

$$A=\underline A_\epsilon\#\frac{1}{\epsilon}(\text{$1$-instanton})\#\frac{1}{\epsilon}(\text{$-1$-instanton}).$$

Multiple-solutions conjecture. For a rather general family of boundary values, multiple solutions to $(\mathcal{D}_{\epsilon})$ should exist in each component $\mathcal{A}_k(A_0)$, for every $k\ne\pm1$, when $\epsilon>0$ is sufficiently small.

The conjecture extends the paper's multiple-solution result to all topological components, including constructions involving a $1$-instanton and a $-1$-instanton in the component $\mathcal{A}_0(A_0)$. Establishing it would require arguments similar to those in the paper, together with longer and more delicate calculations.